Abstract

The notion of multi-scale representation is essential
to many aspects of early visual processing.
This article deals with the axiomatic formulation of
the special type of multi-scale representation known as
scale-space representation.
Specifically, this work is concerned with the problem of how
different choices of basic assumptions (scale-space axioms)
restrict the class of permissible smoothing operations.

A scale-space formulation previously expressed for discrete signals
is adapted to the continuous domain.
The basic assumptions are that the scale-space family should
be generated by convolution with a one-parameter family of
rotationally symmetric smoothing kernels that satisfy a
semi-group structure and obey a causality condition
expressed as a non-enhancement requirement of local extrema.
Under these assumptions, it is shown that the smoothing kernel is uniquely
determined to be a Gaussian.

Relations between this scale scale-space formulation and
recent formulations based on scale invariance are explained
in detail.
Connections are also pointed out to approaches
based on non-uniform smoothing.